Determination of the ionization and dissociation energies of the

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The ionization and dissociation energies of the hydrogen molecule (H2) and its isotopomers (D2 and HD) are bench- mark quantities in molecular quantum ...
THE JOURNAL OF CHEMICAL PHYSICS 132, 154301 共2010兲

Determination of the ionization and dissociation energies of the deuterium molecule „D2… Jinjun Liu,1 Daniel Sprecher,1 Christian Jungen,2 Wim Ubachs,3 and Frédéric Merkt1,a兲 1

Laboratorium für Physikalische Chemie, ETH-Zürich, 8093 Zürich, Switzerland Laboratoire Aimé Cotton, CNRS II, Bâtiment 505, Campus d’Orsay, 91405 Orsay Cedex, France 3 Department of Physics and Astronomy, Laser Centre, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam , The Netherlands 2

共Received 18 December 2009; accepted 9 March 2010; published online 15 April 2010兲 The transition wave numbers from selected rovibrational levels of the EF 1⌺+g 共v = 0兲 state to selected np Rydberg states of ortho- and para-D2 located below the adiabatic ionization threshold have been measured at a precision better than 10−3 cm−1. Adding these wave numbers to the previously determined transition wave numbers from the X 1⌺+g 共v = 0 , N = 0 , 1兲 states to the EF 1⌺+g 共v = 0 , N = 0 , 1兲 states of D2 and to the binding energies of the Rydberg states calculated by multichannel quantum defect theory, the ionization energies of ortho- and para-D2 are determined to be 124 745.394 07共58兲 cm−1 and 124 715.003 77共75兲 cm−1, respectively. After re-evaluation of the dissociation energy of D+2 and using the known ionization energy of D, the dissociation energy of D2 is determined to be 36 748.362 86共68兲 cm−1. This result is more precise than previous experimental results by more than one order of magnitude and is in excellent agreement with the most recent theoretical value 36 748.3633共9兲 cm−1 关K. Piszczatowski, G. Łach, M. Przybytek et al., J. Chem. Theory Comput. 5, 3039 共2009兲兴. The ortho-para separation of D2, i.e., the energy difference between the N = 0 and N = 1 rotational levels of the X 1⌺+g 共v = 0兲 ground state, has been determined to be 59.781 30共95兲 cm−1. © 2010 American Institute of Physics. 关doi:10.1063/1.3374426兴 I. INTRODUCTION

The ionization and dissociation energies of the hydrogen molecule 共H2兲 and its isotopomers 共D2 and HD兲 are benchmark quantities in molecular quantum mechanics.1,2 Recently, we measured the transition wave number from the EF state of ortho-H2 to a selected Rydberg state below the ground state of ortho-H+2 with a precision of 0.000 29 cm−1.3 Combining this measurement with previous experimental and theoretical results for other energy intervals, the ionization and dissociation energies of the hydrogen molecule have been determined to be 124 417.491 13共37兲 and 36 118.069 62共37兲 cm−1, respectively. Most recently, Piszczatowski et al.2 made a new theoretical calculation of the dissociation energy of H2 and D2 by including nonadiabatic, relativistic, and quantum electrodynamics 共QED兲 corrections. The relativistic and QED corrections were obtained at the adiabatic level of theory by including all contributions of the order ␣2 and ␣3 as well as the major 共one-loop兲 ␣4 term, where ␣ is the fine-structure constant. The theoretical value they obtained for the dissociation energy of H2, 36 118.0695共10兲 cm−1, is in excellent agreement with our value. For the D2 molecule, however, Piszczatowski et al. pointed out a small discrepancy 共at the level of two standard deviations of the experimental value兲 between their theoretical value 关36 748.3633共9兲 cm−1兴 and the most recent experimental result 关36 748.343共10兲 cm−1, Ref. 4兴. A new experimental determination of the dissociation energy of D2 with a similar precision to that reached for H2 would therea兲

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fore be desirable to resolve this discrepancy and would provide another quantity with which future theoretical calculations can be compared. We report here on a new determination of the ionization energies of both ortho- and para-D2 关Ei共ortho-D2兲 ⬅ Ei共D2兲 and Ei共para-D2兲兴 from which we derive a new experimental value of the dissociation energy of D2. The ionization energies are obtained in each case as a sum of three energy intervals 共see Fig. 1兲. In the case of ortho-D2, the first energy interval is between the X 1⌺+g 共v = 0 , N = 0兲 and the EF 1⌺+g 共v = 0 , N = 0兲 levels, the transition wave number of which was measured with high precision previously 关99 461.449 08共11兲 cm−1, Ref. 5兴, the second between the EF 1⌺+g 共v = 0 , N = 0兲 and the 29p21共0兲 Rydberg state belonging to a series converging on the X+ 2⌺+g 共v+ = 0 , N+ = 2兲 level of ortho-D+2 , and the third between the 29p21共0兲 Rydberg state and the X+ 2⌺+g 共v+ = 0 , N+ = 0兲 ionic level. In the case of para-D2, the first energy interval is between the X 1⌺+g 共v = 0 , N = 1兲 and the EF 1⌺+g 共v = 0 , N = 1兲 levels 关99 433.716 38共11兲 cm−1, Ref. 5兴, the second between the EF 1⌺+g 共v = 0 , N = 1兲 and the 52p12共0兲 Rydberg state belonging to a series converging on the X+ 2⌺+g 共v+ = 0 , N+ = 1兲 level of para-D+2 , and the third the binding energy of the 52p12共0兲 Rydberg state. The transition wave numbers between the selected rovibrational levels of the EF state and the Rydberg states were measured using a 共2 + 1⬘兲 resonant three-photon excitation sequence followed by delayed pulsed-field ionization 共PFI兲 of the Rydberg states as in our study of H2.3 The binding energies of the Rydberg states were calculated by multichannel quantum defect theory 共MQDT兲. The accuracy

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N+ 1

(9)

0

X + 2Σ +g (υ + = 0)

(7) (3)

np (8) Ei(para-D2)

(4) Ei(D2)≡Ei(ortho-D2)

np (6) (2)

N 2 1 0

EF 1Σ +g (υ = 0)

(5) (1) 2 (10)

1

X 1Σ +g (υ = 0)

0

FIG. 1. Schematic energy level diagram of D2 illustrating the procedure used to determine the ionization energies and the ortho-para separation. The energy intervals are not to scale. The bold arrows show the multiphoton excitation scheme used in the present experiment. The dashed arrows show the binding energies of the Rydberg states. The numerical values of the energy intervals 共1兲–共10兲 are given in Table III. The hyperfine structures of the different levels have not been drawn for clarity 共see text for details兲.

of the MQDT calculations for D2 has been verified independently in a comparison with a high-resolution measurement of the nf Rydberg spectra of D2 by millimeter-wave spectroscopy6 and also with high-resolution laser spectra of the np Rydberg states 共see below兲. The dissociation energy of D2 关D0共D2兲兴 can be derived using the relation 共see Fig. 5 of Ref. 3兲 D0共D2兲 = Ei共D2兲 + D0共D+2 兲 − Ei共D兲,

共1兲

from the dissociation energy of the molecular cation 关D0共D+2 兲兴, which was re-evaluated using available literature data 共see Appendix B兲, and the ionization energy of the atom 关Ei共D兲兴, which is known very accurately.7 In addition, the ortho-para separation of the neutral molecule ⌬E1−0, defined as the energy level separation between the N = 0 and N = 1 rotational energy levels of the X 1⌺+g 共v = 0兲 state of D2, can be derived using the relation 共see Fig. 1兲: + − Ei共para-D2兲, ⌬E1−0 = Ei共ortho-D2兲 + ⌬E1−0

共2兲

+ is the ortho-para separation of the cation, which where ⌬E1−0 is known very accurately from calculations.8

II. EXPERIMENT

The experimental setup is the same as used in our previous study of H2.3 In brief, the third harmonic of a commercial dye laser 共␭ ⬃ 201 nm, bandwidth ⬃0.04 cm−1兲 was used to excite the EF 1⌺+g 共v = 0 , N = 0 , 1兲 ← X 1⌺+g 共v = 0 , N = 0 , 1兲 two-photon transitions. The second harmonic of the near-infrared 共NIR兲 output of a pulsed titanium-doped sapphire 共Ti:Sa兲 amplifier 共␭ ⬃ 396 nm,

bandwidth ⬃20 MHz兲,9,10 seeded by a Ti:Sa cw ring laser, was then used to access np Rydberg states belonging to series converging on the lowest two rotational levels 共N+ = 0 or 1兲 of the X+ 2⌺+g 共v+ = 0兲 state of D+2 from the selected EF 1⌺+g 共v = 0, N = 0 or 1兲 levels. The 396 nm laser beam was split into two components and introduced into the interaction region in a counterpropagating configuration, both components overlapping spatially with the 201 nm beam. The difference between the transition wave numbers measured using each of these two components represents twice the Doppler shift resulting from a possible nonorthogonality between the molecular beam and the 396 nm laser beams and can be eliminated by taking the average value of the two measurements. The transitions were detected by ionizing the Rydberg states created by the two-step process using a pulsed electric field applied with a time delay with respect to the laser pulses. The generated cations were extracted by the same field and accelerated toward a microchannel plate detector. Spectra were obtained by monitoring the D+2 ion signal as a function of the wave number of the second laser. The absolute and relative calibration of the Ti:Sa-cwring-laser frequency was carried out by recording, simultaneously with the PFI spectra, the Doppler-free saturation absorption spectrum of I2 3,5,11 and the transmission signal through a high-finesse Fabry–Pérot etalon locked to a polarization-stabilized He–Ne laser.12 The frequency shift arising in the multipass amplification in the Ti:Sa crystals 共⬍10 MHz兲 was determined by monitoring the interference between the pulse-amplified laser beam with the cw NIR output of the ring laser13 and taking the Fourier transform of the beat-note signal recorded when the 396 nm laser was on resonance with the EF to Rydberg transitions. III. RESULTS A. Survey spectra of Rydberg states and MQDT calculations

In our previous work on H2, the binding energy of the selected 54p11共0兲 Rydberg state of ortho-H2 was taken from the result of a millimeter-wave spectroscopic experiment analyzed by MQDT at submegahertz accuracy.14 In a similar millimeter-wave experiment carried out on para-D2,6 transitions from n = 51d – 53d to n = 54f – 57f Rydberg states belonging to series converging on the ground state of para-D+2 could also be measured and extrapolated to their series limits using the same MQDT parameters as used for ortho-H2 at the same accuracy. These results confirmed the expectation that isotopic substitution does not affect the eigenquantum defects and demonstrated the ability of MQDT to predict the binding energy of Rydberg states with submegahertz accuracy. No millimeter-wave transitions to the np states accessed in the present experiment were observed for para-D2. The binding energies of these np states were therefore calculated by MQDT in the present work, as listed in the second column of Table I. These values correspond to energy level separations between the centers of gravity of the hyperfine structures of both the Rydberg states and the N+ = 0 共N+ = 1兲

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Dissociation energy of D2

TABLE I. Binding energies of the Rydberg states of ortho- D2 共para-D2兲 calculated by MQDT, their experimental wave numbers relative to the 29p21 共52p12兲 states determined from the survey spectra, and the binding energies of the 29p21 共52p12兲 states calculated as sums of these two quantities 共in cm−1兲. Rydberg state nᐉNN+

50p01 29p21b 51p01 52p01 53p01 54p01 55p01

Rydberg state nᐉNN+ 24p32b 50p12 51p12 52p12 53p12 54p12 55p12 56p12 57p12

MQDT binding energy

44.193 51 43.232 89 42.014 44 40.517 04 39.051 17 37.661 24 36.370 53

Experimental wave number relative to 29p21 ortho-D2 ⫺0.961 90共80兲 0 1.217 20共80兲 2.715 88共80兲 4.181 28共80兲 5.571 52共80兲 6.862 34共80兲 Standard deviation

MQDT binding energy

45.104 08 43.666 74 42.096 42 40.552 23 39.073 74 37.669 40 36.339 05 35.081 98 33.898 69

Experimental wave number relative to 52p12 para-D2 ⫺4.550 18共80兲 ⫺3.114 74共80兲 ⫺1.543 32共80兲 0 1.478 74共80兲 2.884 40共80兲 4.213 60共80兲 5.471 70共80兲 6.655 06共80兲 Standard deviation

Experimental binding energy of 29p21a

43.231 61 43.232 89 43.231 64 43.232 92 43.232 45 43.232 76 43.232 87 0.000 58 Experimental binding energy of 52p12a

40.553 90 40.552 00 40.553 10 40.552 23 40.552 48 40.553 80 40.552 65 40.553 68 40.553 75 0.000 74

a Calculated by adding up the MQDT binding energy of each Rydberg state and its relative position 共in cm−1兲 with respect to the 29p21 共52p12兲 state of ortho-D2 共para- D2兲 determined in the survey spectra. b State with the largest contribution from the 29p21 共24p32兲 zero-order state. Because of the strong interaction between the np01 and np21 共np12 and np32兲 channels of ortho-D2 共para-D2兲, all Rydberg states are mixed.

level of ortho-D+2 共para-D+2 兲. The hyperfine structure of the Rydberg states of para-D2 was determined by MQDT in a full calculation of the type described in Refs. 6 and 14 and the hyperfine structure of para-D+2 was taken from Ref. 6. The binding energies were found to be identical to within 1 MHz with the binding energies determined in MQDT calculations neglecting nuclear spins. For ortho-D2 Rydberg states and ortho-D+2 , for which the hyperfine structure is not known, we have therefore determined the binding energy in MQDT calculations neglecting nuclear spins. These are also expected to be accurate within 1 MHz. One should note here that the hyperfine splittings of the np 共S = 0兲 states at n ⬍ 60 is extremely small 共typically less than 5 MHz兲 because the exchange interaction is much larger than the hyperfine interaction.14 In the present calculation, the MQDT parameters are the same as those determined for H2 in Ref. 14 and ionization channels associated with vibrational levels of the cation with v+ ⱕ 9 were included. The energies of the ionic levels were taken from Ref. 8. The p Rydberg states are labeled nᐉNN+ 共S兲 with all quantum numbers having their usual meanings 共see Ref. 14 for details兲. A strong interaction between the N+ = 0 and N+ = 2 channels of ortho-D2 and between the N+ = 1 and N+ = 3 channels of para-D2 leads to the observation, for instance, of the 29p21 level in spectra of the np01 levels. This interaction is perfectly described by the MQDT calculations.

Although it has been proven that the same MQDT parameters can be used to describe the nf levels of both ortho-H2 and para-D2 with submegahertz accuracy,6 we decided to independently verify the accuracy of the calculations of np states of both ortho- and para-D2. For this reason, we recorded the survey spectra of the np Rydberg series of ortho- and para-D2 in the region close to the nd and nf states that were accessed in the previous millimeter-wave experiments.6 The spectra of ortho- and para-D2 are displayed in Figs. 2共a兲 and 2共b兲, respectively, and consist each of one series of strong transitions from the EF 1⌺+g 共v = 0, N = 0 and 1兲 states to np Rydberg states belonging to series converging on the X+ 2⌺+g 共v+ = 0 , N+ = 0兲 level of ortho-D+2 and on the X+ 2⌺+g 共v+ = 0 , N+ = 1兲 level of para-D+2 , respectively. The third column of Table I gives their positions 共in cm−1兲 relative to the 29p21 and the 52p12 states in the case of ortho- and para-D2, respectively. These two states are those that were chosen in the measurements using counterpropagating laser beams to determine the ionization energies 共see Sec. III B兲. Adding up the MQDT binding energy and the relative wave number of each Rydberg state measured experimentally yields independent values of the binding energies of the 29p21 and the 52p12 states, which are listed in the last column of Table I. The uncertainties 共one standard deviation兲 of the binding energies determined in this way are

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(a)

nlN N+ ( S = 0)

+

D2 signal / arb. units

0.0

50 p01 29 p 21

51 p01

52 p01

53 p01

54 p01

55 p01

-1.0 -2.0 -3.0 -4.0 -5.0 -6.0

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

56 p12

57 p12

20

21

-1

wave number/cm -124690 [with respect to X 1Σ +g (υ = 0, N = 0)]

(b)

nlN N+ ( S = 0)

+

D2 signal / arb. units

-0.5

24 p32

50 p12

51 p12

52 p12

53 p12

54 p12

55 p12

-1.0 -1.5 -2.0 -2.5 -3.0

7

8

9

10

11

12

13

14

15

16

17

18

19

22

-1

wave number/cm -124660 [with respect to X 1Σ +g (υ = 0, N = 1)] FIG. 2. PFI spectra of Rydberg states of 共a兲 ortho- D2 and 共b兲 para-D2. The relative intensities are very sensitive to experimental conditions and are not reliable.

0.000 58 cm−1 and 0.000 74 cm−1 for the 29p21 state of ortho-D2 and the 52p12 state of para-D2, respectively, and implicitly include the uncertainties of the MQDT calculation and of the relative frequency calibration of the spectra. The estimated experimental uncertainty is 0.000 80 cm−1, and the standard deviations of the binding energy calculation originates entirely from the experimental uncertainty, which consequently validates the accuracy of the MQDT calculation for the np states: errors in the binding energies calculated by MQDT of the order of 10 MHz would have been detected, and we believe that the MQDT calculations of the np Rydberg states of D2 are as accurate as those of H2, i.e., better than 1 MHz.14 Note that other weaker transitions were also observed and assigned to transitions to other Rydberg states but were not included into the analysis because of their poor signal-to-noise ratio.

trating the calibration of the transition frequency to the 52p12共0兲 state of para-D2 is displayed in Fig. 3. For each transition, ten measurements were carried out on different days, each measurement consisting of two spectra, one recorded with the 396 nm laser beam propagating parallel, and the other with the 396 nm laser propagating antiparallel to the 201 nm laser beam. The fundamental transition wave numbers determined from these measurements are plotted with their uncertainties in Fig. 4. The center wave +

(a) para-D2

FWHM=55 MHz

52p12(0) (b) I2

a2, P181, B-X (0-14) FWHM=3 MHz

B. Measurements using counterpropagating laser beams FSR=149.969(1) MHz

In order to eliminate possible Doppler shifts of the EF to Rydberg transition frequencies and determine the ionization energies with better accuracy, the transition wave numbers from the N = 0 共N = 1兲 levels of the EF 1⌺+g 共v = 0兲 state to the 29p21共0兲 Rydberg state of ortho-D2 关52p12共0兲 state of para-D2兴 were determined in separate high-precision measurements using counterpropagating laser beams. These two transitions were selected because their fundamental wave numbers are in the vicinity of an I2 line, the frequency of which had previously been determined to a precision better than 100 kHz using a frequency comb.5 An example illus-

(c) etalon

12620.16

12620.18

12620.30

12620.32

12620.34

-1

fundamental NIR wave number / cm

FIG. 3. 共a兲 Spectrum of the transition from the EF 1⌺+g 共v = 0 , N = 1兲 level to the 52p12共S = 0兲 level of para-D2. 关共b兲 and 共c兲兴 I2 calibration and etalon traces, respectively. The position of the a2 hyperfine component of the P181, B − X 共0-14兲 rovibronic transition of I2 was determined to be 378 342 844.89共3兲 MHz using a frequency comb 共Ref. 5兲 and used for the absolute frequency calibration.

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fundamental wave number / cm

-1

154301-5

(a)

transition wave numbers from the measurements displayed in Fig. 4 are given as contributions 共2兲 and 共6兲 in Table III. Their uncertainties are taken as the linear summation of the statistical and systematic uncertainties.

12620.3570

12620.3565

12620.3560

D. Ionization energies, dissociation energy, and the ortho-para separation

12620.3555

12620.3550 1

-1

12620.3690

fundamental wave number / cm

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2

3

2

3

4

5

6

7

8

9

10

4

5

6

7

8

9

10

(b)

12620.3685

12620.3680

12620.3675

12620.3670

12620.3665 1

measurement FIG. 4. Distribution of the calibrated fundamental wave numbers of 共a兲 the 29p21共0兲 ← EF 1⌺+g 共v = 0 , N = 0兲 transition of ortho-D2 and 共b兲 the 52p12共0兲 ← EF 1⌺+g 共v = 0 , N = 1兲 transition of para-D2. Triangles and squares represent independent measurements with each of the two counterpropagating 396 nm laser beams. Closed circles are the mean values of pairs of measurements. The dashed lines mark the positions of the overall mean values 关12 620.356 05共11兲 cm−1 and 12 620.367 58共18兲 cm−1 for orthoand para-D2, respectively.兴 The size of the experimental uncertainty is indicated by the vertical bars.

numbers of the transitions 共full circles兲 were obtained by taking the average of both measurements. The frequency difference between the two scans of each pair, i.e., twice the Doppler shift, amounted to ⬃2 ⫻ 50 MHz, and the mean transition wave numbers are 2 ⫻ 12 620.356 05共11兲 cm−1 and 2 ⫻ 12 620.367 58共18兲 cm−1 for ortho- and para-D2, respectively, where the numbers in parentheses represent one standard deviation in the unit of the last digit. The frequency shift in the Ti:Sa amplifier has been corrected for in the transition wave numbers listed above. The larger standard deviation of the para-D2 measurements is attributed to the weaker transition intensities. C. Error budget for the EF to Rydberg transition wave numbers

The same calibration procedure as used in the previous work on H2 共Ref. 3兲 was used for D2 and its details are not repeated in the body of the article but are presented in Appendix A. Here only the resulting error budget is presented in Table II. As in our previous investigation of H2 we evaluated statistical and systematic uncertainties separately. All uncertainties were assumed to be independent of each other and are summarized in Table II. The total statistical and systematic uncertainties were calculated as quadrature summations of all respective contributions. The calibrated EF to Rydberg

Figure 1 illustrates the energy level diagram used in the determination of the ionization energies of ortho- and para-D2. The numerical values of all relevant energy intervals are listed in Table III and lead to values of the ionization energies of ortho- and para-D2 of 124 745.394 07共58兲 cm−1 and 124 715.003 77共75兲 cm−1, respectively. These values correspond to the energy separations between the centers of gravity of the hyperfine structure of the N = 0 共1兲 level of ortho-D2 共para-D2兲 and the N+ = 0 共1兲 level of ortho-D+2 共para-D+2 兲. Using the ionization energy of D2 关Ei共D2兲 ⬅ Ei共ortho-D2兲兴 determined in the present work and the previously calculated values of D0共D+2 兲 −1 关21 711.583 34共35兲 cm , slightly modified from Ref. 8 to take account of higher relativistic and radiative corrections as described in Appendix B兴 and Ei共D兲 关109 708.614 552 99共10兲 cm−1, from Ref. 15兴, the dissociation energy of D2 is determined from Eq. 共1兲 to be 36 748.362 86共68兲 cm−1 共see Table III兲. It is important to note that calculations on one-electron systems are much more accurate than on systems with more than one electron. A possible error in the value of D0共D+2 兲 used here would directly influence the present value of D0共D2兲. The ortho-para separation of D+2 , i.e., the spacing between the N+ = 0 and N+ = 1 levels of the X+ 2⌺+g 共v+ = 0兲 ground state of D+2 , was previously determined by ab initio calculation to be 29.3910共1兲 cm−1.8 Combining this result with the ionization energies of ortho- and para-D2, the orthopara separation of D2 can be calculated using Eq. 共2兲 to be 59.781 30共95兲 cm−1 共see Table III兲. This value is in excellent agreement with, but more accurate than the value 关59.7805共16兲 cm−1兴 which is calculated using the molecular constants previously derived from an electric quadrupole vibration-rotation spectrum of the fundamental vibrational band of D2.16 The overall uncertainties in the ionization energies, the dissociation energy, and the ortho-para separation are determined as quadrature summation of the uncertainties of all energy intervals. The estimated uncertainty of the binding energies of the Rydberg states by MQDT 共1 MHz兲 are based on the previous work on ortho-H2.14 IV. DISCUSSION AND CONCLUSIONS

In the present work, the ionization energies of ortho- and para-D2 have been determined with a precision of 17 and 22 MHz, respectively. These values, derived as combinations of experimentally measured transition frequencies and binding energies of the experimentally accessed Rydberg states calculated based on an MQDT analysis, are confirmed in three different ways:

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TABLE II. Error budget in megahertz. In most cases, individual corrections and errors were determined for the frequency of the cw Ti:Sa laser and are then multiplied by 2. ortho- D2 ⫾3.3⫻ 2

para- D2 ⫾5.4⫻ 2

Statistical uncertainties Uncertainty in the determination of the line centers: I2 spectra D2 spectra Nonlinearity of the ring laser scans Residual Doppler shift Sum in quadrature 共i兲

⫾0.35⫻ 2 ⫾2.8⫻ 2 ⬍ ⫾ 1.1⫻ 2 ⬍ ⫾ 0.23 ⫾6.1

⫾0.37⫻ 2 ⫾3.5⫻ 2 ⬍ ⫾ 1.1⫻ 2 ⬍ ⫾ 0.23 ⫾7.4

Systematic uncertainties Error in linearization resulting from uncertainty of FSR Uncertainty in the position of the I2 reference linesa Frequency shift in the Ti:Sa amplifier Frequency shift in the doubling crystal ac Stark shift by the 396 nm laser Frequency shift by the 201 nm laser dc Stark shift by the stray electric fields Pressure shift Sum in quadrature 共ii兲

⫾0.043⫻ 2 ⬍ ⫾ 0.1⫻ 2 ⫾0.038⫻ 2 ⬍ ⫾ 0.35 ⬍ ⫾ 1.30⫻ 2 ⬍ ⫾ 1.7⫻ 2 ⬍⫾5⫻2 ⬍ ⫾ 0.05 ⫾10.9

⫾0.046⫻ 2 ⬍ ⫾ 0.1⫻ 2 ⫾0.075⫻ 2 ⬍ ⫾ 0.35 ⬍ ⫾ 0.51⫻ 2 ⬍ ⫾ 5.4⫻ 2 ⬍⫾5⫻2 ⬍ ⫾ 0.05 ⫾14.8

⫾17

⫾22

rms of 10 measurements using counterpropagating laser beams

Total uncertainty 关共i兲 + 共ii兲兴 a

Reference 5.

共1兲

共2兲

Survey spectra of np Rydberg states were recorded. The binding energies of selected Rydberg states of ortho-D2 关29p21共0兲兴 and para-D2 关52p12共0兲兴 were determined as sums of the experimental relative transition wave numbers and the MQDT binding energies corresponding to all Rydberg states observed in the spectra 共see Table I兲. The precision of the calculation 共one standard deviation兲 corresponds to the uncertainty of the frequency calibration, which validates the MQDT calculations 共Sec. III A兲. Combining the ionization energies of both ortho- and para-D2 determined in the present work with previously calculated ortho-para separation of D+2 , the ortho-para separation of D2 was determined and found to be con-

共3兲

sistent with the previous experimental value of McKellar and Oka.16 Control measurements of the transition frequencies from the EF state to lower n Rydberg states 共n = 39, binding energy ⬃72 cm−1兲 of both ortho- and para-D2 were also carried out.17 At n = 39 the uncertainties resulting from the stray fields are much reduced, but possible errors in the estimation of the binding energies by MQDT increase. The ionization energies derived as sums of the experimentally measured level energies of the Rydberg states and their binding energies calculated by MQDT agree with those reported in the present paper within the estimated uncertainty. The quantum defects used in the calculations are effective ones, deter-

TABLE III. Energy intervals and determination of the ionization energies, dissociation energy and the ortho-para separation of D2 in cm−1. The labelling of the energy intervals corresponds to Fig. 1.

共1兲 共2兲 共3兲 共4兲 = 共1兲 + 共2兲 + 共3兲 共5兲 共6兲 共7兲 共8兲 = 共5兲 + 共6兲 + 共7兲

共9兲 共10兲 = 共4兲 + 共9兲 − 共8兲

Energy interval

Wave number 共cm−1兲

Reference

EF 1⌺+g 共v = 0 , N = 0兲 — X 1⌺+g 共v = 0 , N = 0兲 29p21共0兲 — EF 1⌺+g 共v = 0 , N = 0兲 + 2 + + X ⌺g 共v = 0 , N+ = 0兲 — 29p21共0兲 共binding energy兲 Ei共D2兲 ⬅ Ei共ortho-D2兲 ⬅ 关X+ 2⌺+g 共v+ = 0 , N+ = 0兲 — X 1⌺+g 共v = 0 , N = 0兲兴 EF 1⌺+g 共v = 0 , N = 1兲 — X 1⌺+g 共v = 0 , N = 1兲 52p12共0兲 — EF 1⌺+g 共v = 0 , N = 1兲 + 2 + + X ⌺g 共v = 0 , N+ = 1兲 — 52p12共0兲 共binding energy兲 Ei共para-D2兲 ⬅ 关X+ 2⌺+g 共v+ = 0 , N+ = 1兲 — X 1⌺+g 共v = 0 , N = 1兲兴 D0共D+2 兲 Ei共D兲 D0共D2兲 = Ei共D2兲 + D0共D+2 兲 − Ei共D兲 X+ 2⌺+g 共v+ = 0 , N+ = 1 — 0兲 共ortho-para separation of D+2 兲 X 1⌺+g 共v = 0 , N = 1 — 0兲 共ortho-para separation of D2兲

99 461.449 08共11兲 25 240.712 10共57兲 43.232 89共3兲 124 745.394 07共58兲 99 433.716 38共11兲 25 240.735 16共74兲 40.552 23共3兲 124 715.003 77共75兲 21 711.583 34共35兲a 109 708.614 552 99共10兲 36 748.362 86共68兲 29.3910共1兲 59.781 30共95兲

5 This work This work This work 5 This work This work This work 8 15 This work 8 This work

Slightly modified to take account of higher relativistic and radiative corrections 共see Appendix B兲.

a

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154301-7

J. Chem. Phys. 132, 154301 共2010兲

Dissociation energy of D2

TABLE IV. Ei共D2兲 as determined in various experimental and theoretical studies or combinations thereof, in cm−1.

0.40

Ei(D2)/cm -124745

1.2

this work

0.38

-1

1.0

Year

0.39

-1

Ei(D2)/cm -124745

0.8

0.37

0.36

0.35 1992

0.6

1996

2000

2004

2008

year

1975 1990 1991 1992 1993 1993 1993 1994 1995 2010

Expt.

Theor.

124 745.6共6兲a 124 745.49共11兲 124 745.382 124 745.353共24兲 124 745.353共24兲 124 745.377 124 745.387 124 745.401 124 745.395 124 745.394 07共58兲

Reference 28 29 30,31 32,33 34 35 36 37 38 This work

a A correction of −1.0 cm−1 was included to account for the pressure shift 共see Ref. 33兲.

0.4

0.2

theoretical experimental

0.0

Our present work on D2 hence serves as an independent verification of the theoretical calculations in Ref. 2. Because the experimental uncertainties in the values of the dissociation energies of H2 and D2 are smaller than those of the best ab initio calculations, they represent a benchmark for future calculations. ACKNOWLEDGMENTS

1975

1980

1985

1990

1995

2000

2005

2010

year FIG. 5. Values for the ionization energy of D2 as determined in various studies or combinations thereof. The theoretical values were reported without uncertainties. The inset represents an enlargement of the area surrounded by the dashed line. The numerical values and the references are listed in Table IV.

mined from fits to experimental data using a theoretical model which does not include doubly excited channels and uses a limited number of vibrational channels. That these approximations do not play a significant role in the present determination has been verified to the best of our experimental capabilities 共see Refs. 6 and 14 and Table I兲. Figure 5 summarizes the results of determinations, by various methods, of the ionization energy of D2 over the past 35 years. The numerical values are also listed in Table IV. Because of the relationship between the dissociation and ionization energies of D2 关Eq. 共1兲兴, the recommended value of the dissociation energy evolved in a very similar manner. The dissociation energy of D2 was determined in the present work as a hybrid experimental-theoretical value. Comparing with the most recent experimental result 关36 748.343共10兲 cm−1, Ref. 4兴, the present value for the dissociation energy of D2 关36 748.362 86共68兲 cm−1兴 is more precise by more than one order of magnitude but differs by two standard deviations of the previous measurement. Our new value is in remarkable agreement with the most recent theoretical value 关36 748.3633共9兲 cm−1, Ref. 2兴. The same excellent agreement was found between our value of the dissociation energy of H2 关36 118.069 62共37兲 cm−1, Ref. 3兴 and the value derived ab initio 关36 118.0695共10兲 cm−1, Ref. 2兴.

This work was financially supported by the European Research Council 共ERC Grant No. 228286兲 and the Swiss National Science Foundation under project No. 200020116245. Additional support from the Swiss Academy of Technical Sciences, Program PHC Germaine de Staël Nr. 2008-35 is also acknowledged. APPENDIX A: EXPERIMENTAL ERRORS AND UNCERTAINTIES

This appendix provides the information on the different contributions to the uncertainties in the determination of the frequencies of the transitions from the EF 1⌺+g 共v = 0 , N = 0 , 1兲 states to the np Rydberg states of ortho- and para-D2 located below the X+ 2⌺+g 共v+ = 0 , N+ = 0 , 1兲 ionization thresholds. 1. Statistical uncertainties

Determination of the line centers. The central wave numbers of the I2 calibration lines were determined by fitting Lorentzian line shapes to the recorded line profiles. The central wave numbers of the EF to Rydberg transitions of D2 were obtained by fitting Gaussian line shapes. The averaged uncertainties 共one standard deviation兲 in determining the center frequencies are 2 ⫻ 0.35 MHz 共2 ⫻ 0.37 MHz兲 and 2 ⫻ 2.8 MHz 共2 ⫻ 3.5 MHz兲 for I2 and D2 transitions in the case of ortho-D2 共para-D2兲, respectively. The larger uncertainty in the case of para-D2 results from the weaker nature of the transitions as discussed in Sec. III B. Nonlinearity of the ring laser. The maximum deviation of the NIR readout frequency of the Ti:Sa ring laser from the actual frequency resulting from the nonlinearity of the scans occurs at the midpoint between two neighboring monitor etalon fringes. From the maximal relative error of ⬃1.5% over

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J. Chem. Phys. 132, 154301 共2010兲

Liu et al.

1 vD ␦ ⌬␯Doppler = · 2 · · ␯0 = 0.23 MHz, 2 c L

共A1兲

where vD2 = 1260 m / s is the speed of D2 in the jet expansion and ␯0 is the transition frequency. The Doppler shift is therefore smaller than 0.23 MHz.

δl=0 dc offset= 2.00 V 0.25 0.20 0.15

1.00 V 0.75 V

0.10

0.50 V 0.05

0.25 V 0V

0.00

-0.25 V -0.50 V

-0.05

-0.75 V -0.10

-1.00 V

+

one free spectral range 共FSR兲 of the monitor etalon, we conclude that the frequency uncertainty resulting from the nonlinearity is always smaller than 共1 / 2兲FSR⫻ 1.5% = 1.1 MHz in the NIR and smaller than 2.2 MHz in the UV. Doppler shift. Although the method of counterpropagating laser beams was employed 共see Sec. III B兲, there still could be a small residual Doppler shift caused by an imperfect alignment of the two 396 nm laser beams. The common beam path of the two beams is L ⬃ 3.5 m. If two beams deviate by ␦ = 0.5 mm 共which is still easily resolvable by eye兲 at the extremity of the common path, the Doppler shift ⌬␯Doppler is

D2 signal / arb. units

154301-8

-0.15

52d12

-0.20

52p12(S=0) -2.00 V

-0.25

2. Systematic errors and uncertainties -0.30

Uncertainty of the FSR. The FSR of the etalon was determined to be 149.9691共13兲 MHz in nine measurements carried out on nine days. Multiplying the uncertainty of the FSR by the number of FSRs separating the I2 line that was used for calibration 关the a2 hyperfine component of the P181, B − X 共0-14兲 rovibronic transition at 12 620.158 873共1兲 cm−1兴 from the position of the Rydberg transitions gives uncertainties of 2 ⫻ 0.043 and 2 ⫻ 0.046 MHz in the transition frequencies of ortho- and para-D2, respectively. I2 calibration by frequency comb. This uncertainty is less than 0.1 MHz in the NIR as explained in Ref. 5. Frequency shifts occurring in the Ti:Sa amplifier. The frequency shifts occurring in the Ti:Sa amplifier, ⫺4.736共38兲 MHz for ortho-D2 measurements and ⫺6.901共75兲 MHz for para-D2 measurements, were determined simultaneously with the measurements described in Sec. III B and have already been corrected for in the transition wave numbers presented in Fig. 4. The larger frequency shift for the para-D2 measurements and its larger uncertainty result from the fact that a higher 396 nm laser power had to be used to compensate for the weaker signal, which could only be achieved by pumping the Ti:Sa crystals with higher Nd:YAG laser intensities. Frequency shifts in the doubling crystal. The frequency shifts arising in frequency doubling have been estimated to be less than 0.35 MHz in our previous study.3 ac stark shifts caused by the 396 nm laser power. To estimate the ac Stark shift, the D2 spectra were recorded at different laser powers and hence different intensities of the 396 nm laser and zero-intensity positions were estimated in a linear extrapolation. The uncertainties associated with the extrapolations are estimated as the product of the slope and the laser powers used in the measurements. Frequency shifts caused by the 201 nm laser power. The transition frequencies were found to be dependent on the 201 nm laser power. When the average laser power is low 共⬍4.5 mW, corresponding to average pulse energies of 180 ␮J; the estimated beam waist in the interaction region is

-0.35

52d13(G=1/2, 3/2)

12620.300

12620.325

12620.350

12620.375

12620.400 -1

fundamental transition wave number / cm

FIG. 6. Spectra of para-D2 recorded in the vicinity of the 52p12共0兲 ← EF 1⌺+g 共v = 0 , N = 1兲 transition with different dc voltage offsets applied across the 7.8-cm-long extraction region. The dotted line indicates the zerofield position of high-ᐉ Rydberg states with ␦ᐉ = 0. The dashed lines indicate the positions of the extreme red- and blueshifted Stark levels 共see text兲. The spectra have been shifted along the vertical axis so that the origin of their intensity scale corresponds to the value of the applied dc electric field in V/cm.

⬃1 mm2兲, the transition frequencies remain constant, but they start increasing at laser powers beyond 4.5 mW. Because we also observed these frequency shifts in experiments in which the 396 nm laser was delayed with respect to the 201 nm laser, they cannot arise from an ac Stark shift induced by the 201 nm laser. Instead, we attribute them to the stray electric fields caused by the ions generated by the 201 nm laser in a 共2 + 1兲 resonance-enhanced multiphoton ionization process. The measurements using counterpropagating laser beams presented in Fig. 4 were carried out at low 201 nm laser power 共⬃1.5 mW for ortho-D2 and ⬃2.0 mW for para-D2兲 so that no frequency corrections needed to be made. Nevertheless, it was necessary to include additional uncertainties of 3.5 MHz for ortho-D2 and 10.8 MHz for para-D2, determined in the same way as for the ac Stark shift caused by the 396 nm laser power. dc Stark shifts caused by stray electric fields. The effects of dc electric fields of different magnitudes and polarities were investigated by applying dc voltages across the 7.8-cmlong stack of cylindrical plates surrounding the ionization region. Figure 6 shows the effect of these dc voltages on the spectrum of para-D2. The transition to the 52d12 state is always observable and its intensity increases with increasing electric field. The linewidth increases as well, presumably

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Reference

Nonrel. energy

7

⫺109 677.583 41

E共H兲

7

⫺109 707.426 59

26, 27

SAV

0.000 04

0.270 68

⫺1.460 95

⫺131 056.875 75

⫺1.599 55

0.000 24

0.270 89 0.350 83

E共HD+兲

26, 27

⫺131 223.436 26

25 Deviations E共D+2 兲

39

⫺1.602 10

0.351 60

⫺1.601 52

0.351 58

⫺0.000 58

0.000 02

⫺1.604 82

0.352 48

⫺0.000 58共30兲a

0.000 02共10兲a

Estimated corrections

⫺1.605 40共32兲b 24

21 379.292 28

0.1379

7, 26, 27

21 379.292 34

0.138 60

21 516.0096

0.1406

7, 26, 27

21 516.009 67

0.141 14

8

21 711.521 05

0.1439

21 711.521 12d

0.144 45共32兲

D0共HD+兲

25

=E共D兲 − E共HD 兲 +

D0共D+2 兲 Estimated D0共D+2 兲 a

+

0.000 31共5兲c

0.000 04

0.001 94

⫺0.000 11

0.000 10

0.002 49

⫺0.000 15

⫺1.187 96 ⫺1.190 04 ⫺1.246 23 ⫺1.248 07

0.000 07

0.002 50

⫺0.000 15

⫺1.247 89 ⫺1.249 99

0.352 50共14兲b

⫺1.252 32 0.000 05共3兲c

0.002 50共3兲c

⫺0.000 15共3兲c

⫺0.0801 ⫺0.000 01

⫺0.080 15 ⫺0.080 71

0.000 06

⫺0.000 56

0.000 04

H+2

⫺0.081 61共14兲

0.057 90 0.0600

⫺0.000 03

⫺0.000 56

0.000 04

⫺0.0816 ⫺0.000 07共5兲

⫺1.250 19共35兲 0.057 82

⫺0.000 02

⫺0.0807

Taken to be the same as for HD but including an uncertainty larger than the difference between the Sum of the value from Ref. 8 and the estimated deviation. c Extrapolated linearly from the values of H+2 and HD+. d Determined using the value of E共D+2 兲 from Ref. 39 and the most recent value of E共D兲 共Ref. 7兲. b

⫺1.188 33 ⫺1.190 26

0.000 07 0.000 18

Estimated deviations

=E共H兲 − E共H+2 兲

⫺0.000 12

total correction

⫺131 418.947 71

8

D0共H+2 兲

0.001 94

R ⬁␣ 5

0.350 76

⫺0.000 73

Deviations

0.000 08

R ⬁␣ 4

0.270 88 0.000 05

⫺1.598 82

24

TP

0.270 66

⫺1.460 92

25 E0共H+2 兲

⫺1.460 95

NS

⫺1.460 92

24 E共D兲

RR

R ⬁␣ 3

Dissociation energy of D2

R ⬁␣ 2

154301-9

TABLE V. Re-evaluation of the dissociation energy of D+2 including all relativistic and radiative corrections up to order R⬁␣4 and the leading two terms of order R⬁␣5. In this treatment one assumes that the perturbation series is convergent. The estimation is based on calculations of Moss 共Refs. 8, 24, 25, and 39兲 on H+2 , HD+, and D+2 , of Korobov 共Refs. 26 and 27兲 on H+2 and HD+, and of CODATA 共Ref. 7兲 on H and D. The upper part of the table gives the ground state energy of the respective species and their corrections. All values are in units of cm−1. The correction terms are listed in Refs. 26 and 27. The corrections are: RR= relativistic and recoil, NS= nuclear size, SAV= self-energy, anomalous magnetic moment, and vacuum polarization, and TP= transverse photon exchange.

0.059 93 0.062 25

⫺0.000 01共3兲

⫺0.000 57共3兲

0.000 04共3兲

0.062 22共35兲

+

and HD values.

J. Chem. Phys. 132, 154301 共2010兲

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154301-10

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Liu et al.

because of Stark splittings and mixing with Stark manifolds of high-ᐉ states. Because of the ⌬ᐉ = ⫾ 1 selection rule of ᐉ-mixing by electric fields, the 52p state mixes first with the 52d state, and then with high-ᐉ states via the 52d state. The zero-field positions of high-ᐉ states 共with quantum defect ␦ᐉ = 0兲 and the extreme red- and blueshifted Stark levels are indicated by the dotted and dashed lines in Fig. 6. The Stark shifts of these levels are calculated using the formula ⌬En = ⫾ 共3 / 2兲Fn2 共in atomic units兲, where F is the electric field. At a dc voltage of 2 V, the 52p and 52d levels are completely immersed in the high-ᐉ manifold of Stark states and the Stark spectra are dominated by a broad asymmetric line. Two new peaks appear at dc offsets beyond 0.25 V and can be assigned to the transitions from the EF state to the 52d13共G = 1 / 2兲 and 52d13共G = 3 / 2兲 Rydberg states based on the results of a previous laser experiment.18 For transitions to np levels of both ortho- and para-D2, no dependence of the transition frequencies on the electric field was observable at low electric fields. 关A quadratic frequency shift with respect to the offset voltage19,20 was observed for the transitions to the 52d13共G = 1 / 2 , 3 / 2兲 states.兴 Consequently, no correction was needed for the dc Stark shift, but an uncertainty of 2 ⫻ 5 MHz= 10 MHz was included to account for the shift at a field of ⫾250 mV/7.8 cm. It is, however, certain that the stray electric field present during the measurements using counterpropagating laser beams summarized in Fig. 4 was smaller. Pressure shift. Herzberg and Jungen21 determined the magnitude of the pressure shift for high-n Rydberg states of H2 to be 5.7⫾ 0.5 cm−1 / amagat, which is expected to be the same for D2. From the estimate of ⬃1013 cm−3 for the D2 number density, we determine the pressure shift to be less than +0.1 MHz with an uncertainty of ⫾0.05 MHz. This pressure shift thus makes a close to negligible contribution to the error budget. APPENDIX B: RE-EVALUATION OF THE DISSOCIATION ENERGY OF D2+

In our derivation of the dissociation energy of D2 we rely on the dissociation energy of D+2 关see Eq. 共1兲兴. This quantity has been calculated in 1993 by Moss to be 21 711.5833 cm−1.8 In his calculations, Moss included relativistic corrections of order R⬁␣2 共taken from Ref. 22兲 and radiative corrections of order R⬁␣3 共using the Bethe logarithm evaluated in Ref. 23兲. In similar calculations he also determined the dissociation energies of other rovibronic states of H+2 , HD+, and D+2 , derived transition frequencies and compared his results with available experimental data.8,24,25 The agreement was in most cases better than the estimated experimental uncertainty of 0.001 cm−1. A re-evaluation of the dissociation energy of D+2 was carried out based on the recent work of Korobov on H+2 and HD+, in which relativistic and radiative corrections of order up to R⬁␣4 共including also the leading R⬁␣5 terms兲 were calculated.26,27 The results of this reevaluation are summarized in Table V, where relativistic and radiative corrections to the energies of H, D, H+2 , HD+, and D+2 , and the dissociation energies of H+2 , HD+, and D+2 are given. While the radia-

tive corrections of order R⬁␣3 are in good agreement with those used by Moss,24,25 there is a discrepancy of −0.0007 cm−1 and −0.0005 cm−1 in the sum of the relativistic and the recoil corrections of order R⬁␣2 in the dissociation energies of H+2 and HD+, respectively. This discrepancy is almost fully compensated by the radiative corrections of order R⬁␣4,26,27 leading to an excellent but coincidental agreement in the value of the overall correction. Based on the corrections needed for H+2 and HD+, we estimated the corrections for the dissociation energy of D+2 as detailed in Table V. The resulting dissociation energy for D+2 is 21 711.583 34共35兲 cm−1, which is almost identical to the value reported by Moss.8 As for H+2 and HD+ the agreement results from an almost exact but coincidental compensation. The rotational energy level structure of the vibronic ground state of D+2 was also taken from Ref. 8 and used in the MQDT calculations 共see Sec. III A兲 and in the determination of the ortho-para separation of D2 关see Eq. 共2兲兴. Relativistic and radiative corrections are almost an order of magnitude smaller for the rotational energies than they are for the dissociation energy so that the values of Moss24,25 and Korobov26,27 for H+2 and HD+ are in perfect agreement. The positions of the rotational energy levels of D+2 were therefore taken without change from Ref. 8. H. Primas and U. Müller-Herold, Elementare Quantenchemie 共Teubner Studienbücher, Stuttgart, 1984兲. Section 5.3 共“Fakten und Zahlen: Die Geschichte des Wasserstoff-Moleküls”兲 gives a complete account of the early efforts invested in the quantitative comparison of experimental and theoretical values of the dissociation energy of H2 and explains in detail how studies of molecular hydrogen contributed to establish the validity of molecular quantum mechanics and to understand chemical bonds physically. 2 K. Piszczatowski, G. Łach, M. Przybytek, J. Komasa, K. Pachucki, and B. Jeziorski, J. Chem. Theory Comput. 5, 3039 共2009兲. 3 J. Liu, E. J. Salumbides, U. Hollenstein, J. C. J. Koelemeij, K. S. E. Eikema, W. Ubachs, and F. Merkt, J. Chem. Phys. 130, 174306 共2009兲. 4 Y. P. Zhang, C. H. Cheng, J. T. Kim, J. Stanojevic, and E. E. Eyler, Phys. Rev. Lett. 92, 203003 共2004兲. 5 S. Hannemann, E. J. Salumbides, S. Witte, R. T. Zinkstok, E.-J. van Duijn, K. S. E. Eikema, and W. Ubachs, Phys. Rev. A 74, 062514 共2006兲. 6 H. A. Cruse, Ch. Jungen, and F. Merkt, Phys. Rev. A 77, 042502 共2008兲. 7 P. J. Mohr, B. N. Taylor, and D. B. Newell, Rev. Mod. Phys. 80, 633 共2008兲. 8 R. E. Moss, J. Chem. Soc., Faraday Trans. 89, 3851 共1993兲. 9 R. Seiler, Th. Paul, M. Andrist, and F. Merkt, Rev. Sci. Instrum. 76, 103103 共2005兲. 10 Th. A. Paul and F. Merkt, J. Phys. B 38, 4145 共2005兲. 11 H. Knöckel, B. Bodermann, and E. Tiemann, Eur. Phys. J. D 28, 199 共2004兲. 12 Th. A. Paul, J. Liu, and F. Merkt, Phys. Rev. A 79, 022505 共2009兲. 13 I. Reinhard, M. Gabrysch, B. F. von Weikersthal, K. Jungmann, and G. zu Putlitz, Appl. Phys. B: Lasers Opt. 63, 467 共1996兲. 14 A. Osterwalder, A. Wüest, F. Merkt, and Ch. Jungen, J. Chem. Phys. 121, 11810 共2004兲. 15 See: http://physics.nist.gov/hdel 共The energy levels of hydrogen and deuterium given in this database include all corrections detailed in Ref. 7. The fundamental constants used in the calculations are taken from CODATA 2002兲. 16 A. R. W. McKellar and T. Oka, Can. J. Phys. 56, 1315 共1978兲. 17 D. Sprecher, J. Liu, and F. Merkt 共unpublished兲. 18 Th. A. Paul, H. A. Cruse, H. J. Wörner, and F. Merkt, Mol. Phys. 105, 871 共2007兲. 19 M. Schäfer and F. Merkt, Phys. Rev. A 74, 062506 共2006兲. 20 A. Osterwalder and F. Merkt, Phys. Rev. Lett. 82, 1831 共1999兲. 21 G. Herzberg and Ch. Jungen, J. Mol. Spectrosc. 41, 425 共1972兲. 22 M. H. Howells and R. A. Kennedy, J. Chem. Soc., Faraday Trans. 86, 3495 共1990兲. 1

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J. Chem. Phys. 132, 154301 共2010兲

Dissociation energy of D2

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